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Theorem mulsrpr 9500
Description: Multiplication of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.)
Assertion
Ref Expression
mulsrpr  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( [ <. A ,  B >. ]  ~R  .R  [ <. C ,  D >. ]  ~R  )  =  [ <. (
( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  )

Proof of Theorem mulsrpr
Dummy variables  x  y  z  w  v  u  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opelxpi 4866 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  -> 
<. A ,  B >.  e.  ( P.  X.  P. ) )
2 enrex 9491 . . . . 5  |-  ~R  e.  _V
32ecelqsi 7419 . . . 4  |-  ( <. A ,  B >.  e.  ( P.  X.  P. )  ->  [ <. A ,  B >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  )
)
41, 3syl 17 . . 3  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  [ <. A ,  B >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  ) )
5 opelxpi 4866 . . . 4  |-  ( ( C  e.  P.  /\  D  e.  P. )  -> 
<. C ,  D >.  e.  ( P.  X.  P. ) )
62ecelqsi 7419 . . . 4  |-  ( <. C ,  D >.  e.  ( P.  X.  P. )  ->  [ <. C ,  D >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  )
)
75, 6syl 17 . . 3  |-  ( ( C  e.  P.  /\  D  e.  P. )  ->  [ <. C ,  D >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  ) )
84, 7anim12i 570 . 2  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( [ <. A ,  B >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  [ <. C ,  D >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  ) ) )
9 eqid 2451 . . . 4  |-  [ <. A ,  B >. ]  ~R  =  [ <. A ,  B >. ]  ~R
10 eqid 2451 . . . 4  |-  [ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R
119, 10pm3.2i 457 . . 3  |-  ( [
<. A ,  B >. ]  ~R  =  [ <. A ,  B >. ]  ~R  /\ 
[ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  )
12 eqid 2451 . . 3  |-  [ <. ( ( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  =  [ <. ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C
) ) >. ]  ~R
13 opeq12 4168 . . . . . . . . 9  |-  ( ( w  =  A  /\  v  =  B )  -> 
<. w ,  v >.  =  <. A ,  B >. )
1413eceq1d 7400 . . . . . . . 8  |-  ( ( w  =  A  /\  v  =  B )  ->  [ <. w ,  v
>. ]  ~R  =  [ <. A ,  B >. ]  ~R  )
1514eqeq2d 2461 . . . . . . 7  |-  ( ( w  =  A  /\  v  =  B )  ->  ( [ <. A ,  B >. ]  ~R  =  [ <. w ,  v
>. ]  ~R  <->  [ <. A ,  B >. ]  ~R  =  [ <. A ,  B >. ]  ~R  ) )
1615anbi1d 711 . . . . . 6  |-  ( ( w  =  A  /\  v  =  B )  ->  ( ( [ <. A ,  B >. ]  ~R  =  [ <. w ,  v
>. ]  ~R  /\  [ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  ) 
<->  ( [ <. A ,  B >. ]  ~R  =  [ <. A ,  B >. ]  ~R  /\  [ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  ) ) )
17 simpl 459 . . . . . . . . . . 11  |-  ( ( w  =  A  /\  v  =  B )  ->  w  =  A )
1817oveq1d 6305 . . . . . . . . . 10  |-  ( ( w  =  A  /\  v  =  B )  ->  ( w  .P.  C
)  =  ( A  .P.  C ) )
19 simpr 463 . . . . . . . . . . 11  |-  ( ( w  =  A  /\  v  =  B )  ->  v  =  B )
2019oveq1d 6305 . . . . . . . . . 10  |-  ( ( w  =  A  /\  v  =  B )  ->  ( v  .P.  D
)  =  ( B  .P.  D ) )
2118, 20oveq12d 6308 . . . . . . . . 9  |-  ( ( w  =  A  /\  v  =  B )  ->  ( ( w  .P.  C )  +P.  ( v  .P.  D ) )  =  ( ( A  .P.  C )  +P.  ( B  .P.  D
) ) )
2217oveq1d 6305 . . . . . . . . . 10  |-  ( ( w  =  A  /\  v  =  B )  ->  ( w  .P.  D
)  =  ( A  .P.  D ) )
2319oveq1d 6305 . . . . . . . . . 10  |-  ( ( w  =  A  /\  v  =  B )  ->  ( v  .P.  C
)  =  ( B  .P.  C ) )
2422, 23oveq12d 6308 . . . . . . . . 9  |-  ( ( w  =  A  /\  v  =  B )  ->  ( ( w  .P.  D )  +P.  ( v  .P.  C ) )  =  ( ( A  .P.  D )  +P.  ( B  .P.  C
) ) )
2521, 24opeq12d 4174 . . . . . . . 8  |-  ( ( w  =  A  /\  v  =  B )  -> 
<. ( ( w  .P.  C )  +P.  ( v  .P.  D ) ) ,  ( ( w  .P.  D )  +P.  ( v  .P.  C
) ) >.  =  <. ( ( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. )
2625eceq1d 7400 . . . . . . 7  |-  ( ( w  =  A  /\  v  =  B )  ->  [ <. ( ( w  .P.  C )  +P.  ( v  .P.  D
) ) ,  ( ( w  .P.  D
)  +P.  ( v  .P.  C ) ) >. ]  ~R  =  [ <. ( ( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  )
2726eqeq2d 2461 . . . . . 6  |-  ( ( w  =  A  /\  v  =  B )  ->  ( [ <. (
( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  =  [ <. ( ( w  .P.  C )  +P.  ( v  .P.  D ) ) ,  ( ( w  .P.  D )  +P.  ( v  .P.  C
) ) >. ]  ~R  <->  [
<. ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C
) ) >. ]  ~R  =  [ <. ( ( A  .P.  C )  +P.  ( B  .P.  D
) ) ,  ( ( A  .P.  D
)  +P.  ( B  .P.  C ) ) >. ]  ~R  ) )
2816, 27anbi12d 717 . . . . 5  |-  ( ( w  =  A  /\  v  =  B )  ->  ( ( ( [
<. A ,  B >. ]  ~R  =  [ <. w ,  v >. ]  ~R  /\ 
[ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  )  /\  [ <. ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C
) ) >. ]  ~R  =  [ <. ( ( w  .P.  C )  +P.  ( v  .P.  D
) ) ,  ( ( w  .P.  D
)  +P.  ( v  .P.  C ) ) >. ]  ~R  )  <->  ( ( [ <. A ,  B >. ]  ~R  =  [ <. A ,  B >. ]  ~R  /\  [ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  )  /\  [
<. ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C
) ) >. ]  ~R  =  [ <. ( ( A  .P.  C )  +P.  ( B  .P.  D
) ) ,  ( ( A  .P.  D
)  +P.  ( B  .P.  C ) ) >. ]  ~R  ) ) )
2928spc2egv 3136 . . . 4  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( ( [
<. A ,  B >. ]  ~R  =  [ <. A ,  B >. ]  ~R  /\ 
[ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  )  /\  [ <. ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C
) ) >. ]  ~R  =  [ <. ( ( A  .P.  C )  +P.  ( B  .P.  D
) ) ,  ( ( A  .P.  D
)  +P.  ( B  .P.  C ) ) >. ]  ~R  )  ->  E. w E. v ( ( [
<. A ,  B >. ]  ~R  =  [ <. w ,  v >. ]  ~R  /\ 
[ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  )  /\  [ <. ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C
) ) >. ]  ~R  =  [ <. ( ( w  .P.  C )  +P.  ( v  .P.  D
) ) ,  ( ( w  .P.  D
)  +P.  ( v  .P.  C ) ) >. ]  ~R  ) ) )
30 opeq12 4168 . . . . . . . . . 10  |-  ( ( u  =  C  /\  t  =  D )  -> 
<. u ,  t >.  =  <. C ,  D >. )
3130eceq1d 7400 . . . . . . . . 9  |-  ( ( u  =  C  /\  t  =  D )  ->  [ <. u ,  t
>. ]  ~R  =  [ <. C ,  D >. ]  ~R  )
3231eqeq2d 2461 . . . . . . . 8  |-  ( ( u  =  C  /\  t  =  D )  ->  ( [ <. C ,  D >. ]  ~R  =  [ <. u ,  t
>. ]  ~R  <->  [ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  ) )
3332anbi2d 710 . . . . . . 7  |-  ( ( u  =  C  /\  t  =  D )  ->  ( ( [ <. A ,  B >. ]  ~R  =  [ <. w ,  v
>. ]  ~R  /\  [ <. C ,  D >. ]  ~R  =  [ <. u ,  t >. ]  ~R  ) 
<->  ( [ <. A ,  B >. ]  ~R  =  [ <. w ,  v
>. ]  ~R  /\  [ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  ) ) )
34 simpl 459 . . . . . . . . . . . 12  |-  ( ( u  =  C  /\  t  =  D )  ->  u  =  C )
3534oveq2d 6306 . . . . . . . . . . 11  |-  ( ( u  =  C  /\  t  =  D )  ->  ( w  .P.  u
)  =  ( w  .P.  C ) )
36 simpr 463 . . . . . . . . . . . 12  |-  ( ( u  =  C  /\  t  =  D )  ->  t  =  D )
3736oveq2d 6306 . . . . . . . . . . 11  |-  ( ( u  =  C  /\  t  =  D )  ->  ( v  .P.  t
)  =  ( v  .P.  D ) )
3835, 37oveq12d 6308 . . . . . . . . . 10  |-  ( ( u  =  C  /\  t  =  D )  ->  ( ( w  .P.  u )  +P.  (
v  .P.  t )
)  =  ( ( w  .P.  C )  +P.  ( v  .P. 
D ) ) )
3936oveq2d 6306 . . . . . . . . . . 11  |-  ( ( u  =  C  /\  t  =  D )  ->  ( w  .P.  t
)  =  ( w  .P.  D ) )
4034oveq2d 6306 . . . . . . . . . . 11  |-  ( ( u  =  C  /\  t  =  D )  ->  ( v  .P.  u
)  =  ( v  .P.  C ) )
4139, 40oveq12d 6308 . . . . . . . . . 10  |-  ( ( u  =  C  /\  t  =  D )  ->  ( ( w  .P.  t )  +P.  (
v  .P.  u )
)  =  ( ( w  .P.  D )  +P.  ( v  .P. 
C ) ) )
4238, 41opeq12d 4174 . . . . . . . . 9  |-  ( ( u  =  C  /\  t  =  D )  -> 
<. ( ( w  .P.  u )  +P.  (
v  .P.  t )
) ,  ( ( w  .P.  t )  +P.  ( v  .P.  u ) ) >.  =  <. ( ( w  .P.  C )  +P.  ( v  .P.  D
) ) ,  ( ( w  .P.  D
)  +P.  ( v  .P.  C ) ) >.
)
4342eceq1d 7400 . . . . . . . 8  |-  ( ( u  =  C  /\  t  =  D )  ->  [ <. ( ( w  .P.  u )  +P.  ( v  .P.  t
) ) ,  ( ( w  .P.  t
)  +P.  ( v  .P.  u ) ) >. ]  ~R  =  [ <. ( ( w  .P.  C
)  +P.  ( v  .P.  D ) ) ,  ( ( w  .P.  D )  +P.  ( v  .P.  C ) )
>. ]  ~R  )
4443eqeq2d 2461 . . . . . . 7  |-  ( ( u  =  C  /\  t  =  D )  ->  ( [ <. (
( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  =  [ <. ( ( w  .P.  u )  +P.  (
v  .P.  t )
) ,  ( ( w  .P.  t )  +P.  ( v  .P.  u ) ) >. ]  ~R  <->  [ <. ( ( A  .P.  C )  +P.  ( B  .P.  D
) ) ,  ( ( A  .P.  D
)  +P.  ( B  .P.  C ) ) >. ]  ~R  =  [ <. ( ( w  .P.  C
)  +P.  ( v  .P.  D ) ) ,  ( ( w  .P.  D )  +P.  ( v  .P.  C ) )
>. ]  ~R  ) )
4533, 44anbi12d 717 . . . . . 6  |-  ( ( u  =  C  /\  t  =  D )  ->  ( ( ( [
<. A ,  B >. ]  ~R  =  [ <. w ,  v >. ]  ~R  /\ 
[ <. C ,  D >. ]  ~R  =  [ <. u ,  t >. ]  ~R  )  /\  [ <. ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C
) ) >. ]  ~R  =  [ <. ( ( w  .P.  u )  +P.  ( v  .P.  t
) ) ,  ( ( w  .P.  t
)  +P.  ( v  .P.  u ) ) >. ]  ~R  )  <->  ( ( [ <. A ,  B >. ]  ~R  =  [ <. w ,  v >. ]  ~R  /\  [ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  )  /\  [
<. ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C
) ) >. ]  ~R  =  [ <. ( ( w  .P.  C )  +P.  ( v  .P.  D
) ) ,  ( ( w  .P.  D
)  +P.  ( v  .P.  C ) ) >. ]  ~R  ) ) )
4645spc2egv 3136 . . . . 5  |-  ( ( C  e.  P.  /\  D  e.  P. )  ->  ( ( ( [
<. A ,  B >. ]  ~R  =  [ <. w ,  v >. ]  ~R  /\ 
[ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  )  /\  [ <. ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C
) ) >. ]  ~R  =  [ <. ( ( w  .P.  C )  +P.  ( v  .P.  D
) ) ,  ( ( w  .P.  D
)  +P.  ( v  .P.  C ) ) >. ]  ~R  )  ->  E. u E. t ( ( [
<. A ,  B >. ]  ~R  =  [ <. w ,  v >. ]  ~R  /\ 
[ <. C ,  D >. ]  ~R  =  [ <. u ,  t >. ]  ~R  )  /\  [ <. ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C
) ) >. ]  ~R  =  [ <. ( ( w  .P.  u )  +P.  ( v  .P.  t
) ) ,  ( ( w  .P.  t
)  +P.  ( v  .P.  u ) ) >. ]  ~R  ) ) )
47462eximdv 1766 . . . 4  |-  ( ( C  e.  P.  /\  D  e.  P. )  ->  ( E. w E. v ( ( [
<. A ,  B >. ]  ~R  =  [ <. w ,  v >. ]  ~R  /\ 
[ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  )  /\  [ <. ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C
) ) >. ]  ~R  =  [ <. ( ( w  .P.  C )  +P.  ( v  .P.  D
) ) ,  ( ( w  .P.  D
)  +P.  ( v  .P.  C ) ) >. ]  ~R  )  ->  E. w E. v E. u E. t ( ( [
<. A ,  B >. ]  ~R  =  [ <. w ,  v >. ]  ~R  /\ 
[ <. C ,  D >. ]  ~R  =  [ <. u ,  t >. ]  ~R  )  /\  [ <. ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C
) ) >. ]  ~R  =  [ <. ( ( w  .P.  u )  +P.  ( v  .P.  t
) ) ,  ( ( w  .P.  t
)  +P.  ( v  .P.  u ) ) >. ]  ~R  ) ) )
4829, 47sylan9 663 . . 3  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( (
( [ <. A ,  B >. ]  ~R  =  [ <. A ,  B >. ]  ~R  /\  [ <. C ,  D >. ]  ~R  =  [ <. C ,  D >. ]  ~R  )  /\  [ <. (
( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  =  [ <. ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C
) ) >. ]  ~R  )  ->  E. w E. v E. u E. t ( ( [ <. A ,  B >. ]  ~R  =  [ <. w ,  v
>. ]  ~R  /\  [ <. C ,  D >. ]  ~R  =  [ <. u ,  t >. ]  ~R  )  /\  [ <. (
( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  =  [ <. ( ( w  .P.  u )  +P.  (
v  .P.  t )
) ,  ( ( w  .P.  t )  +P.  ( v  .P.  u ) ) >. ]  ~R  ) ) )
4911, 12, 48mp2ani 684 . 2  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  E. w E. v E. u E. t ( ( [
<. A ,  B >. ]  ~R  =  [ <. w ,  v >. ]  ~R  /\ 
[ <. C ,  D >. ]  ~R  =  [ <. u ,  t >. ]  ~R  )  /\  [ <. ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C
) ) >. ]  ~R  =  [ <. ( ( w  .P.  u )  +P.  ( v  .P.  t
) ) ,  ( ( w  .P.  t
)  +P.  ( v  .P.  u ) ) >. ]  ~R  ) )
50 ecexg 7367 . . . 4  |-  (  ~R  e.  _V  ->  [ <. (
( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  e.  _V )
512, 50ax-mp 5 . . 3  |-  [ <. ( ( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  e.  _V
52 simp1 1008 . . . . . . . 8  |-  ( ( x  =  [ <. A ,  B >. ]  ~R  /\  y  =  [ <. C ,  D >. ]  ~R  /\  z  =  [ <. ( ( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  )  ->  x  =  [ <. A ,  B >. ]  ~R  )
5352eqeq1d 2453 . . . . . . 7  |-  ( ( x  =  [ <. A ,  B >. ]  ~R  /\  y  =  [ <. C ,  D >. ]  ~R  /\  z  =  [ <. ( ( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  )  -> 
( x  =  [ <. w ,  v >. ]  ~R  <->  [ <. A ,  B >. ]  ~R  =  [ <. w ,  v >. ]  ~R  ) )
54 simp2 1009 . . . . . . . 8  |-  ( ( x  =  [ <. A ,  B >. ]  ~R  /\  y  =  [ <. C ,  D >. ]  ~R  /\  z  =  [ <. ( ( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  )  -> 
y  =  [ <. C ,  D >. ]  ~R  )
5554eqeq1d 2453 . . . . . . 7  |-  ( ( x  =  [ <. A ,  B >. ]  ~R  /\  y  =  [ <. C ,  D >. ]  ~R  /\  z  =  [ <. ( ( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  )  -> 
( y  =  [ <. u ,  t >. ]  ~R  <->  [ <. C ,  D >. ]  ~R  =  [ <. u ,  t >. ]  ~R  ) )
5653, 55anbi12d 717 . . . . . 6  |-  ( ( x  =  [ <. A ,  B >. ]  ~R  /\  y  =  [ <. C ,  D >. ]  ~R  /\  z  =  [ <. ( ( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  )  -> 
( ( x  =  [ <. w ,  v
>. ]  ~R  /\  y  =  [ <. u ,  t
>. ]  ~R  )  <->  ( [ <. A ,  B >. ]  ~R  =  [ <. w ,  v >. ]  ~R  /\ 
[ <. C ,  D >. ]  ~R  =  [ <. u ,  t >. ]  ~R  ) ) )
57 simp3 1010 . . . . . . 7  |-  ( ( x  =  [ <. A ,  B >. ]  ~R  /\  y  =  [ <. C ,  D >. ]  ~R  /\  z  =  [ <. ( ( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  )  -> 
z  =  [ <. ( ( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  )
5857eqeq1d 2453 . . . . . 6  |-  ( ( x  =  [ <. A ,  B >. ]  ~R  /\  y  =  [ <. C ,  D >. ]  ~R  /\  z  =  [ <. ( ( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  )  -> 
( z  =  [ <. ( ( w  .P.  u )  +P.  (
v  .P.  t )
) ,  ( ( w  .P.  t )  +P.  ( v  .P.  u ) ) >. ]  ~R  <->  [ <. ( ( A  .P.  C )  +P.  ( B  .P.  D
) ) ,  ( ( A  .P.  D
)  +P.  ( B  .P.  C ) ) >. ]  ~R  =  [ <. ( ( w  .P.  u
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )
)
5956, 58anbi12d 717 . . . . 5  |-  ( ( x  =  [ <. A ,  B >. ]  ~R  /\  y  =  [ <. C ,  D >. ]  ~R  /\  z  =  [ <. ( ( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  )  -> 
( ( ( x  =  [ <. w ,  v >. ]  ~R  /\  y  =  [ <. u ,  t >. ]  ~R  )  /\  z  =  [ <. ( ( w  .P.  u )  +P.  (
v  .P.  t )
) ,  ( ( w  .P.  t )  +P.  ( v  .P.  u ) ) >. ]  ~R  )  <->  ( ( [ <. A ,  B >. ]  ~R  =  [ <. w ,  v >. ]  ~R  /\  [ <. C ,  D >. ]  ~R  =  [ <. u ,  t
>. ]  ~R  )  /\  [
<. ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C
) ) >. ]  ~R  =  [ <. ( ( w  .P.  u )  +P.  ( v  .P.  t
) ) ,  ( ( w  .P.  t
)  +P.  ( v  .P.  u ) ) >. ]  ~R  ) ) )
60594exbidv 1772 . . . 4  |-  ( ( x  =  [ <. A ,  B >. ]  ~R  /\  y  =  [ <. C ,  D >. ]  ~R  /\  z  =  [ <. ( ( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  )  -> 
( E. w E. v E. u E. t
( ( x  =  [ <. w ,  v
>. ]  ~R  /\  y  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
( w  .P.  u
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )  <->  E. w E. v E. u E. t ( ( [ <. A ,  B >. ]  ~R  =  [ <. w ,  v
>. ]  ~R  /\  [ <. C ,  D >. ]  ~R  =  [ <. u ,  t >. ]  ~R  )  /\  [ <. (
( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  =  [ <. ( ( w  .P.  u )  +P.  (
v  .P.  t )
) ,  ( ( w  .P.  t )  +P.  ( v  .P.  u ) ) >. ]  ~R  ) ) )
61 mulsrmo 9498 . . . 4  |-  ( ( x  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  y  e.  ( ( P.  X.  P. ) /.  ~R  )
)  ->  E* z E. w E. v E. u E. t ( ( x  =  [ <. w ,  v >. ]  ~R  /\  y  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
( w  .P.  u
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )
)
62 df-mr 9483 . . . . 5  |-  .R  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e. 
R.  /\  y  e.  R. )  /\  E. w E. v E. u E. t ( ( x  =  [ <. w ,  v >. ]  ~R  /\  y  =  [ <. u ,  t >. ]  ~R  )  /\  z  =  [ <. ( ( w  .P.  u )  +P.  (
v  .P.  t )
) ,  ( ( w  .P.  t )  +P.  ( v  .P.  u ) ) >. ]  ~R  ) ) }
63 df-nr 9481 . . . . . . . . 9  |-  R.  =  ( ( P.  X.  P. ) /.  ~R  )
6463eleq2i 2521 . . . . . . . 8  |-  ( x  e.  R.  <->  x  e.  ( ( P.  X.  P. ) /.  ~R  )
)
6563eleq2i 2521 . . . . . . . 8  |-  ( y  e.  R.  <->  y  e.  ( ( P.  X.  P. ) /.  ~R  )
)
6664, 65anbi12i 703 . . . . . . 7  |-  ( ( x  e.  R.  /\  y  e.  R. )  <->  ( x  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  y  e.  ( ( P.  X.  P. ) /.  ~R  )
) )
6766anbi1i 701 . . . . . 6  |-  ( ( ( x  e.  R.  /\  y  e.  R. )  /\  E. w E. v E. u E. t ( ( x  =  [ <. w ,  v >. ]  ~R  /\  y  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
( w  .P.  u
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )
)  <->  ( ( x  e.  ( ( P. 
X.  P. ) /.  ~R  )  /\  y  e.  ( ( P.  X.  P. ) /.  ~R  ) )  /\  E. w E. v E. u E. t
( ( x  =  [ <. w ,  v
>. ]  ~R  /\  y  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
( w  .P.  u
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )
) )
6867oprabbii 6346 . . . . 5  |-  { <. <.
x ,  y >. ,  z >.  |  ( ( x  e.  R.  /\  y  e.  R. )  /\  E. w E. v E. u E. t ( ( x  =  [ <. w ,  v >. ]  ~R  /\  y  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
( w  .P.  u
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )
) }  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  y  e.  (
( P.  X.  P. ) /.  ~R  ) )  /\  E. w E. v E. u E. t
( ( x  =  [ <. w ,  v
>. ]  ~R  /\  y  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
( w  .P.  u
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )
) }
6962, 68eqtri 2473 . . . 4  |-  .R  =  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  y  e.  (
( P.  X.  P. ) /.  ~R  ) )  /\  E. w E. v E. u E. t
( ( x  =  [ <. w ,  v
>. ]  ~R  /\  y  =  [ <. u ,  t
>. ]  ~R  )  /\  z  =  [ <. (
( w  .P.  u
)  +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  (
v  .P.  u )
) >. ]  ~R  )
) }
7060, 61, 69ovig 6418 . . 3  |-  ( ( [ <. A ,  B >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  [
<. C ,  D >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  [ <. ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C
) ) >. ]  ~R  e.  _V )  ->  ( E. w E. v E. u E. t ( ( [ <. A ,  B >. ]  ~R  =  [ <. w ,  v
>. ]  ~R  /\  [ <. C ,  D >. ]  ~R  =  [ <. u ,  t >. ]  ~R  )  /\  [ <. (
( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  =  [ <. ( ( w  .P.  u )  +P.  (
v  .P.  t )
) ,  ( ( w  .P.  t )  +P.  ( v  .P.  u ) ) >. ]  ~R  )  ->  ( [ <. A ,  B >. ]  ~R  .R  [ <. C ,  D >. ]  ~R  )  =  [ <. ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C
) ) >. ]  ~R  ) )
7151, 70mp3an3 1353 . 2  |-  ( ( [ <. A ,  B >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  [
<. C ,  D >. ]  ~R  e.  ( ( P.  X.  P. ) /.  ~R  ) )  -> 
( E. w E. v E. u E. t
( ( [ <. A ,  B >. ]  ~R  =  [ <. w ,  v
>. ]  ~R  /\  [ <. C ,  D >. ]  ~R  =  [ <. u ,  t >. ]  ~R  )  /\  [ <. (
( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  =  [ <. ( ( w  .P.  u )  +P.  (
v  .P.  t )
) ,  ( ( w  .P.  t )  +P.  ( v  .P.  u ) ) >. ]  ~R  )  ->  ( [ <. A ,  B >. ]  ~R  .R  [ <. C ,  D >. ]  ~R  )  =  [ <. ( ( A  .P.  C )  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C
) ) >. ]  ~R  ) )
728, 49, 71sylc 62 1  |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
)  ->  ( [ <. A ,  B >. ]  ~R  .R  [ <. C ,  D >. ]  ~R  )  =  [ <. (
( A  .P.  C
)  +P.  ( B  .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B  .P.  C ) )
>. ]  ~R  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    /\ w3a 985    = wceq 1444   E.wex 1663    e. wcel 1887   _Vcvv 3045   <.cop 3974    X. cxp 4832  (class class class)co 6290   {coprab 6291   [cec 7361   /.cqs 7362   P.cnp 9284    +P. cpp 9286    .P. cmp 9287    ~R cer 9289   R.cnr 9290    .R cmr 9295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-oadd 7186  df-omul 7187  df-er 7363  df-ec 7365  df-qs 7369  df-ni 9297  df-pli 9298  df-mi 9299  df-lti 9300  df-plpq 9333  df-mpq 9334  df-ltpq 9335  df-enq 9336  df-nq 9337  df-erq 9338  df-plq 9339  df-mq 9340  df-1nq 9341  df-rq 9342  df-ltnq 9343  df-np 9406  df-plp 9408  df-mp 9409  df-ltp 9410  df-enr 9480  df-nr 9481  df-mr 9483
This theorem is referenced by:  mulclsr  9508  mulcomsr  9513  mulasssr  9514  distrsr  9515  m1m1sr  9517  1idsr  9522  00sr  9523  recexsrlem  9527  mulgt0sr  9529
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